Understanding the properties of a generic state given by the superposition of different spacetimes configuration is one of the major questions that quantum gravity aim to address. This has major implication for the physics of the early universe, where the seeds of cosmic structures are originated from quantum fluctuations of the geometry. Predicting the strength of correlations of geometrical observables in this regime gives direct access to the initial conditions of our universe, from which all matter structures later evolved. Exploiting techniques from covariant Loop Quantum Gravity, I present an operational recipe to construct physical states of the quantum geometry, to define a cosmological interpretation for them, and compute analytically and numerically the quantities that characterize them, i.e. correlations between the volumes of spacially-separated region and the corresponding entanglement entropy. I review the results obtained and I discuss the future perspective of this research program.